Can You Mathematically Transform Sin(x) to Y=X?

[Tido611]

is there a way to turn a sin curve (sin(x)) from a horizontal graph to a 45 degree, y=x graph mathematically

 

[James R]

Yes.

Your initial equation is y=sin x

Put

x = x' \cos \theta - y' \sin \theta
y = x' \sin \theta + y' \cos \theta

where, in your case \theta = 45 degrees.

Then, your new equation is:

\frac{1}{\sqrt{2}}(x' + y') = \sin \left[ \frac{1}{\sqrt{2}}(x' - y') \right]

Now, the only problem you have is to rearrange this so as to write y' in terms of x'.

 

[TD]

You rotated over an angle of -45° because your minus-sign was placed wrongly. You have to be careful whether it's the coordinate axis which are being rotated or the function itself. In this case, the rotation matrix is:

\left( {\begin{array}{*{20}c}<br /> {\cos \theta } &amp; {\sin \theta } \\<br /> { - \sin \theta } &amp; {\cos \theta } \\<br /> \end{array}} \right)

giving the new equation:

\frac{1}{\sqrt{2}}(y&#039; - x&#039;) = \sin \left[ \frac{1}{\sqrt{2}}(x&#039; + y&#039;) \right]

 

[James R]

Fair enough.